Notes on geometry
Author(s)
Bibliographic Information
Notes on geometry
（Universitext）
Springer-Verlag, 1988, c1983
Corr. 2nd print. [ed.]
- : us
- : gw
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Bibliography: p. 105-109
Includes index
Description and Table of Contents
Description
In recent years, geometry has played a lesser role in undergraduate courses than it has ever done. Nevertheless, it still plays a leading role in mathematics at a higher level. Its central role in the history of mathematics has never been disputed. It is important, therefore, to introduce some geometry into university syllabuses. There are several ways of doing this, it can be incorporated into existing courses that are primarily devoted to other topics, it can be taught at a first year level or it can be taught in higher level courses devoted to differential geometry or to more classical topics. These notes are intended to fill a rather obvious gap in the literature. It treats the classical topics of Euclidean, projective and hyperbolic geometry but uses the material commonly taught to undergraduates: linear algebra, group theory, metric spaces and complex analysis. The notes are based on a course whose aim was two fold, firstly, to introduce the students to some geometry and secondly to deepen their understanding of topics that they have already met. What is required from the earlier material is a familiarity with the main ideas, specific topics that are used are usually redone.
Table of Contents
I: Euclidean Geometry.- The Linear Groups.- The Relationship Between O(n) and GL(n,R).- Affine Subspaces and Affine Independence.- Isometries of Rn.- Isometries of R2.- Isometries of R3.- Some Subsets of R3.- Finite Groups of Isometries.- The Platonic Solids.- Duality.- The Symmetry Groups of the Platonic Solids.- Finite Groups of Rotations of R3.- Crystals.- Rotations and Quaternions.- Problems.- II: Projective Geometry.- Homogeneous Co-ordinates.- The Topology of P1 and P2.- Duality.- Projective Groups.- The Cross-Ratio.- Fixed Points of Projectivities.- The Elliptic Plane.- Conics.- Diagonalization of Quadratic Forms.- Polarity.- Problems.- III: Hyperbolic Geometry.- The Parallel Axiom.- The Beltrami (or projective) Model.- Stereographic Projection.- The Poincare Model.- The Local Metric.- Areas.- Trigonometry.- Hyperbolic Trigonometry.- Lines and Polarity.- Isometries.- Elliptic Trigonometry.- Problems.- Further Reading.- List of Symbols.
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